A domain-theoretic approach to fuzzy metric spaces

We introduce a partial order ⊑ M on the set B X of formal balls of a fuzzy metric space ( X , M , ∧ ) in the sense of Kramosil and Michalek, and discuss some of its properties. We also characterize when the poset ( B X , ⊑ M ) is a continuous domain by means of a new notion of fuzzy metric completeness introduced here. The well-known theorem of Edalat and Heckmann that a metric space is complete if and only if its poset of formal balls is a continuous domain, is deduced from our characterization.